Functions for estimating aboveground biomass of birch in Norway

A suite of regional allometric aboveground biomass functions were derived for Betula pubescens and Betula pendula for Norwegian conditions. The data consisted of 67 trees sampled throughout Norway. A total of 14 component functions were developed for total aboveground, total stem, stemwood, stem bark, live crown, live branch, leaf, and dead branch biomass using combinations of diameter at breast height and height as predictor variables. Application of the derived functions to existing local southern Norwegian mountain birch and regional Swedish biomass datasets indicated an overall good predictive ability of the developed functions. However, the functions produced slight underestimates, suggesting that the respective birch populations had differing biomass allocation patterns. When the developed functions were applied to Norwegian National Forest Inventory data, they produced slightly higher biomass stock and stock change estimates than what is obtained using existing Swedish functions. The higher estimates were evident in the north, central, and western part of Norway, while estimates were similar in southeastern Norway where growing conditions are most similar to Swedish conditions. The analysis indicates that the derived functions are the best available for regional birch biomass stock and stock change estimation in Norway.


Introduction
Tree biomass stock and stock change estimation are central for forest-based bioenergy feedstock assessments, in studies of the terrestrial carbon cycle, and for reporting under the United Nations Framework Convention on Climate Change (UNFCCC) and the Kyoto Protocol. Currently, Norway uses the regional Swedish functions developed by Marklund (1987Marklund ( , 1988 for reporting Picea abies (L.) Karst (Norway spruce), Pinus sylvestris L. (Scots pine), Betula pubescens Ehrh. (downy birch), and Betula pendula Roth (silver birch) biomass. The practice of geographically extrapolating the functions to Norway likely leads to some unknown error in the biomass estimate resulting from the potentially differing biomass allocation patterns of the same tree species growing in different conditions. B. pendula and B. pubescens with its high elevation subspecies B. pubescens Ehrh. ssp. czerepanóvii (N.I. Orlova) Hämet-Ahti (mountain birch) are the two main birch species in Norway. Downy birch is the most common of the two birch species, comprising over 95% of total birch volume and occurring throughout the country (Norwegian National Forest Inventory [NNFI] 2009). Silver birch occurs at lower elevations and has a more southerly distribution, although the species is also locally present in parts of northern Norway up to a latitude of 66°N and in eastern parts of Finnmark county (NNFI 2009). Birch is the third most common tree genus in Norway, representing 16% of the standing tree volume behind Scots pine (30%) and Norway spruce (45%) (Granhus et al. 2012).
Several local biomass functions have been derived for both birch species in Norway, but they are limited in their applicable geographic range. Functions for southeastern silver birch were developed for stem, branch, and leaf biomass by Korsmo (1995). Mountain birch functions have been derived from birch sampled from the southwest (Kjelvik 1974;Bollandsås et al. 2009) and the southeast (Opdahl 1987;Bollandsås et al. 2009) for various biomass components including aboveground, stem, stemwood, stem bark, total crown, branches, and leaves. The existing local functions do not cover large areas of the birch zone in Norway including low elevation coastal western, midelevation southeastern, central, or northern Norway.
Birch is one of the most common trees by growing stock throughout Scandinavia, the Baltic nations, Russia, and locally in northern North America (Global Forest Resources Assessment [GFRA] 2010). Regional allometric birch biomass functions have been developed for Iceland (Snorrason & Einarsson 2006), Sweden (Marklund 1987(Marklund , 1988, and Finland (Repola 2008). Regional generalized regression (Pastor et al. 1984) functions have been developed for birch from meta-analyses of existing local functions for Canada (Lambert et al. 2005;Ung et al. 2008) and the USA (Jenkins et al. 2003). To the authors' knowledge, no regional birch functions currently exist for the Baltic nations or Russia.
The objectives of this study were to: (1) derive regional allometric aboveground biomass functions for birch in Norway; (2) compare the derived functions by applying them to two existing birch biomass datasets; and (3) use NNFI data to compare the birch biomass stock and stock change estimates obtained with the derived functions with estimates from existing local southern Norwegian mountain birch functions and the regional Swedish functions currently used to obtain birch biomass stock and stock change estimates for Norway.

Site and sample tree selection
In order to obtain empirical allometric functions for biomass estimation, individual birch trees were destructively sampled. Sample site locations were subjectively selected to represent (to the extent possible) the range of conditions in which birch occurs in Norway. Sample site selection was initially made by dividing Norway into four regions: southeastern, western, central, and northern. For each region, four to five sites were located ( Figure 1) and within each site, four trees were sampled with adequate spacing from each other, resulting in a total of 67 sampled trees on 17 sites (one small tree was lost during processing). Within each region, the sample sites were located to represent the regional variability in site, stand, and tree variables ( Table 1). The sample trees were selected from vigorous rot-free trees to reflect the full diameter at breast height at 1.3 m (dbh) range present on the site. For each sample tree, a 250 m 2 (r = 8.92 m) plot was established with the sample tree as plot center. Species and dbh were recorded for all trees (other than birch) on the plot with a total height in excess of 50% of the dominant tree height in young Figure 1. Birch biomass sampling site locations. Seventeen total sites were selected with five located in the southeast, four in the west, four in central, and four in northern Norway. Table 1. Descriptive data for the current study, 9th NNFI, Marklund (1987Marklund ( , 1988, and Bollandsås et al. (2009)  ; crown height = distance from the ground to the base of the live crown (ignoring one time a single live branch if separated by more than two whorls from the next live branch); basal area = stand basal area; birch site index = the dominant height of the largest tree by dbh at the reference age of 40 years at breast height (Strand 1967); elevation = meters above sea level; plot proportion birch (%) = percentage of birch stems within sample tree plot (r = 8.92 m); diameter-to-height-ratio = dbh (cm)/height (m). Temperature (Tveito et al. 2000) and precipitation (Tveito et al. 1997) data are derived from climate data for all of Norway; normal from 1961 to 1990. NNFI birch site index = the mean height of the 100 largest trees by dbh at the reference age of 40 years at breast height per hectare.
stands or with a dbh > 5 cm in older stands (Supplementary material, Appendix A). No differentiation between downy and silver birch species was made for the sample trees due to the phenotypic plasticity of identifying traits between the two species that vary with growing conditions and age (Atkinson 1992;Atkinson et al. 1997). All sampled trees were growing on mineral soils with depth between 15 cm and greater than 70 cm.

Destructive sampling
The sample trees were felled and cross-cut at stump height after measuring the total tree height and height-tolive crown (Supplementary material, Appendix A) from the stump surface. Branch sampling was then carried out mostly following the methodology of Marklund (1987Marklund ( , 1988 and all weighing was done with tripod-suspended field scales (OCS™, 500 kg, ±0.1 kg for large pieces or UWE™, HS-15K, ±0.01 kg for smaller pieces). First, an estimate of the total number of live branches in the live crown was obtained by: (1) dividing the live crown into three equal lengths, (2) counting the number of live branches present in a centered 1.3 m subsection for each crown part, and (3) summing the branch counts from each section. Second, live sample branches were randomly sampled by delimbing the crown starting from the base of the live crown according to: live sample branch = (estimate of the total number of live branches / 5) × (random number between 0 and 1). After each sampled branch was cut and set aside for further processing, the formula was applied again until between five and nine live sample branches were sampled per tree depending on the results of iteratively applying the formula. The fresh weight (FW) of live sample branches was recorded with leaves and catkins (if present) attached using a portable table-top scale (UWE™, SHC-6C, ±0.2 g). Then, woody branch material, leaves, and catkins (if present) of the live sample branches were separated and packaged for storage. The FW of the live crown was obtained by summing the FW of all live sample branches and remaining live branches with leaves and catkins (if present) attached. All dead branches (if present) were also weighed to obtain the total FW of dead branches and a subjectively selected sample (ca. 500 g) was brought to the lab after FW determination in the field. On the delimbed stem, distance from marked dbh to stump height, and stem diameter starting from 0.5 m below dbh to a 5 cm top was recorded in 0.5 m intervals.
Stem disk sampling was performed by dividing the stem into eight (if dbh ≥ 7 cm) or four (if dbh < 7 cm) sections of equal length from the stump surface to the tip. Disk locations were randomly selected within each stem section. A disk was taken at each location and an additional disk was taken at breast height (1.3 m above mean ground level). The disks were approximately 2-4 cm thick when the stem diameter was greater than 7 cm and 20-40 cm when the stem diameter was smaller than 7 cm. The distance from the stem disk to the stump height was measured. Stem disk FW was taken with bark intact and cross-sectional over-bark and under-bark diameters were recorded in two perpendicular directions. Total stem FW was determined by cutting up the remaining stem into sections and weighing them with a field scale (OCS™) attached to a portable tripod and adding the total sample disk FW to the obtained weight.
All sampled materials were placed in paper bags and as soon as logistically possible (typically 0-2 days), placed either in a dry ventilated room (ca. 20°C) or cold dry storage (<0°C) (depending on availability) before being sent to the lab for further processing.

Lab work and data compilation
In order to obtain dry weight (DW) of the samples, all sample materials were divided into smaller pieces (except stem disks which were left intact) and placed in paper bags. All samples were placed in a forced-air oven at 103°C for 2-11 days depending on sample size, and dried until minimal daily relative mass loss was achieved. The bark was removed from the stem disks after drying and weighed separately.
Age at breast height was determined from the dbh disk by counting the year rings under a stereo microscope. The site index was calculated for each sample site from using the height of the tree with the biggest dbh and its age (Strand 1967). Sample site values for mean annual temperature (Tveito et al. 2000), minimum and maximum monthly mean temperature (Tveito et al. 2000), and mean annual precipitation (Tveito et al. 1997) were projected into a Geographic Information System (Quantum GIS 1.8.0-Lisboa) and obtained for the current study, Marklund (1987Marklund ( , 1988 and Bollandsås et al. (2009) datasets.

Aboveground biomass dataset
The sampled field and lab data were combined to construct estimates for the following eight biomass components for each tree: total stem, stemwood, stem bark, live crown (live branches, leaves, and catkins if present), live branch, leaf, dead branch, and total aboveground biomass. Various methodologies used in the field and lab phases of the project made it necessary to employ a specific multistage process in order to construct the component biomass estimates for each tree. The important steps of the process are outlined here; a more detailed description is available in Appendix B along with other Supplementary material.
Total stem biomass was determined by calculating: (1) the DW to FW ratio for each stem disk and assigning each disk to the appropriate stem section; (2) the volumes of each stem section using Smalian's formula; (3) the total stem volume as the sum of the stem section volumes, with forked tree volumes being calculated in the same way for each forked and single stem; (4) the volume-weighted DW to FW ratio of the stem; and (5) the volume-weighted total stem biomass (Supplementary material, Appendices A and B).
Stemwood biomass was determined by calculating: (6) the cross-sectional area of the over-bark and stemwood portions of each sample disk; (7) the proportion of stemwood cross-sectional area of each disk assigned to the corresponding stem section; (8) the proportion of the total stem volume that the stem section represents; (9) the proportion of the stemwood in each stem section; (10) the volume-weighted proportion of stemwood in the stem; and (11) the volume-weighted stemwood biomass (Supplementary material, Appendix B).
Stem bark biomass was determined by calculating: (12) the proportion of stem bark for each sample disk assigned to the corresponding section; (13) the proportion of the stem bark in the section; (14) the volumeweighted proportion of stem bark in the tree; and (15) the volume-weighted stem bark biomass. Live crown biomass was determined by calculating: (16) the sum of the DWs of woody branches, leaves, and catkins for each sample branch; (17)

Function development
Single-and two-variable nonlinear mixed-effects (NLME) functions were fit to the component biomass data in order to account for the data's inherent hierarchical, nonlinear, and heteroscedastic structure (Parresol 1999(Parresol , 2001. Linearizing log transformations (Baskerville 1972) of the response and/or predictor variables were not performed in order to avoid potential bias problems associated with back transformation to the original scale from linearized function fits (Flewelling & Pienaar 1981;Duan 1983;Taylor 1986;Wirth et al. 2004;Wutzler et al. 2008). All functions were fit and evaluated following the NLME procedures outlined in Pinheiro and Bates (2000) and Robinson and Hamann (2010) with the NLME package (Pinheiro et al. 2012) available in R statistical software (R Core Team 2012). All fixed and random effects function assumptions and best fits were evaluated at each function development stage with a combination of diagnostic plots and lowest Akaike information criterion (AIC) value. The best function across all single-and two-variable functions was selected by the lowest rootmean-square error (RMSE).
Single-variable functions with dbh as the sole predictor were derived for total aboveground (TAG d ), total stem (TS d ), stemwood (SW d ), stem bark (SB d ), live crown (LC d ), live branch (LB d ), leaf (LF d ), and dead branch (DB d ) biomass. The best function form was initially determined by visually fitting scatterplots (Bates & Watts 1988;Sit & Costello 1994) of each of the biomass components against dbh as a predictor. As previously found by many authors (Parresol 1999;Lambert et al. 2005;Johansson 2007;Wutzler et al. 2008), the power function (Sit & Costello 1994) best represented all component biomass data (Equations (1) and (3)): where Y js is the observed biomass of tree j at site s, X jds is the observed value for tree j of explanatory variable d (dbh) at site s, β o and β d are parameters to be estimated for the fixed effects, α ds represents the random effects for the variable d on site s, and ε js are the residuals. Sample site-wise random effects α ds were only assigned to the β d parameter for all functions.
A "power of covariate" variance function (Equation (2)) was used to model the variance structure of the withinsite errors for all functions (Pinheiro & Bates 2000).
where n js is the absolute value of the variance covariate and δ is an unrestricted parameter allowing for cases where variance increases or decreases with n js (Pinheiro & Bates 2000). For all component biomass fractions, functions were fit using Equations (1) and (2) with equal variance weights of 1 and a variance covariate given by the fitted values (default value) except in the stem bark function (SB d ) where a fixed value of δ = 0.9 was used because the NLME with the default value would not converge (i.e. could not be fit).
The random effects and variance function (Equation (2)) are implicitly part of, but are not explicitly stated in, the final single-variable function as they only reflect sitelevel deviations from the fixed effects. Two-variable functions were derived with dbh and height as predictors for total aboveground (TAG dh ), total stem (TS dh ), stemwood (SW dh ), stem bark (SB dh ), live crown (LC dh ), and live branch (LB dh ) biomass. Prior to the choice of including height as a second variable, nine other candidate predictor variables were evaluated for inclusion in the single-variable component biomass functions including: age at dbh, crown length (data not shown), average crown width (data not shown), crown height, site index, plot basal area around the sample tree (data not shown), stems per hectare, elevation, and region (data not shown) ( Table 1). The candidate variables were assessed by a series of tests: (1) visual assessment of scatterplots of the standardized residuals of the single-variable biomass component functions against predictor values (Bates & Watts 1988); (2) statistical assessment using a Bonferroni corrected twosided t-test of function standardized residuals stratified into subjective low, medium, and high categories (α = 0.05); (3) statistical test for a significant (α = 0.05) trend of linear function fits of the function residuals against predictor variable values. Based on these evaluations, height was the preferred second variable.
The form of the function for the two-variable functions was determined in the same way as the single-variable functions except that both dbh and height were used in conjunction as the predictors. The inclusion of height additively, multiplicatively, and with different curve forms (Sit & Costello 1994) was evaluated. NLME function fits were attempted for all tested function forms. The best function form for all two-variable functions is as follows: where X jhs is the observed value of explanatory variable h (height) for tree j at site s and β h is a parameter to be estimated for the fixed effects. As in the single-variable functions, sample site-wise random effects were only assigned to β d and the variance structure of the within-site errors was modeled with Equation (2) using the default values. The final twovariable function (Equation (3)) does not explicitly state the random effects or the variance function as in the single-variable function.
The possibility of developing a set of three-variable functions with each of the remaining nine predictor variables was evaluated, but upon careful consideration of the variable selection tests, the relative importance of the variables to improve individual tree biomass estimation, and the general availability of the variables in inventory data, no such functions were derived.
In order to keep the modeling approach as simple as possible, the nonlinear seemingly unrelated regression process (Parresol 2001) used to force the true additivety of component functions for total aboveground biomass was not performed. Therefore, no across-model contemporaneous correlations (Parresol 2001) are accounted for in any total aboveground estimation presented here. The derived total aboveground biomass combinations (TAG dh , TAG combination1 = TS dh + LC dh + DB d , and TAG combination2 = SW dh + SB d + LB dh + LF d + DB d ) had mean predicted differences of 0.9%, 0.6%, and 0.1%, respectively, compared to the same data used to derive the functions.

Comparing the functions with existing data
In order to test if the derived functions were correctly specified, they were applied to two existing datasets. The first was a local southern Norwegian mountain birch dataset (Bollandsås et al. 2009) and the second a regional Swedish birch dataset inventoried in the mid-1980s (Marklund 1987(Marklund , 1988. Three function evaluation metrics were calculated: (1) RMSE for the prediction errors; (2) t-test of the mean of the prediction errors; and (3) linear function fit of the prediction errors over predictor variables to check for trends. For the Norwegian function comparison, observed biomass values for total aboveground (stem + total crown) (Supplementary material, Appendix A), stem, and total crown were used; predicted values were calculated using the derived functions for total aboveground (TAG dh ), total stem (TS dh ), and complete crown (live crown (LC dh ) + dead branches (DB d )) (Supplementary material, Appendix A). Prediction errors were plotted against dbh. For the function comparison against Swedish data, observed stemwood, stem bark, live branch, and dead branch biomass values were compared with the predictions of the derived functions for stemwood (SW dh ), stem bark (SB d ), live branch (LB dh ), and dead branch (DB d ) biomass. Prediction errors were plotted against dbh, height, age, site index, and elevation (Marklund 1987(Marklund , 1988. Leaf biomass was not included in the Swedish function comparison as no birch leaf biomass was available from Marklund (1987Marklund ( , 1988.

Norwegian birch biomass stock and stock change estimates
To explore whether the new and existing birch biomass functions differ markedly with respect to their predictions of birch biomass stock and stock change from NNFI data in Norway, estimates were calculated with the following function combinations: (1) total aboveground birch biomass (TAG S ) = SW dh + SB d + LB dh + DB d + LF d (Tables 2 and 3) (current study); (2) TAG M = stemwood (B-5) + stem bark (B-8) + live branch (B-11) + dead branch (B-16) (Marklund 1987(Marklund , 1988) + leaves (where leaf biomass = B-5 × (0.011 a /0.52 b ) ( a factor currently applied by NNFI for UNFCCC reporting; b de Wit et al. 2006); (3) TAG B = total stem ("Stem") + total crown ("Tree crown") biomass (Bollandsås et al. 2009). Marklund's (1987Marklund's ( , 1988 function combination used here is currently used for regional birch biomass estimation in Norway. Calculations were made on data from two consecutive 5-year and DB d , dead branch biomass functions. "XX d " represents the biomass function "XX" fit with only dbh (subscript "d") as the predictor. β o and β d are parameter estimates for the fixed effects. N = number of observations. α ds = sample site-wise random effects only assigned to the β d parameter. δ = estimated power value of the variance structure model.
h . TAG dh , total aboveground; TS dh , total stem; SW dh , stemwood; SB dh , stem bark; LC dh , live crown; and LB dh , live branch biomass functions. "XX dh " represents the biomass function "XX" fit with dbh (subscript "d") and height (subscript "h") as the predictors. β o , β d , and β h are parameter estimates for the fixed effects. α ds = sample site-wise random effects only assigned to the β d parameter. δ = estimated power value of the variance structure model. a Best biomass function by RMSE between the single-and two-variable functions in the current study. inventory cycles of the NNFI, restricted to: (1) undivided plots on forestry land with birch present and (2) plots that were inventoried during  and available for re-measurement during 2005-2009 (NNFI9) (N = 6353 plots, plot radii = 8.92 m). All birch used in the calculation had measured dbh. On plots with 10 trees or less, all the heights were measured, while for plots with more than 10 trees, a relascope-selected subsample with a target of 10 trees per plot was measured. The remaining tree heights on the plot were modeled following the standard tariff approach applied by NNFI (see Antón-Fernández & Astrup 2012 and references therein). It can be expected that stock and stock change estimates using modeled heights are less variable compared to those using measured heights; however, since the same trees were used for all estimates, all comparisons were equivalent.
The regional biomass estimates were further grouped into Norwegian regions by Norwegian county groups according to: southeast = Oppland, Buskerud, Vestfold, Hedmark, Oslo, Akershus, Østfold, Telemark, Aust-Agder, and Vest-Agder; west = Møre og Romsdal, Sogn og Fjordane, Hordaland, and Rogaland; central = Nord-Trøndelag and Sør Trøndelag; and north = Finnmark, Troms, and Nordland counties. Calculated estimates were also grouped by site productivity classes which are grouped Norwegian site indices for birch according to: unproductive = potential yield < 1 m 3 ha −1 yr −1 ; low = height at 40 years of age (H40) 6-8 m; medium = H40 11-14 m; high = H40 17-23 m (Strand 1967). Calculated estimates were finally grouped by forest types which were defined as: birch dominant = plots with ≥ 70% composition of birch; other deciduous = plots with ≥ 70% composition of deciduous trees in total (birch < 70%); mixed forest = other mixed stand types; conifer dominant = plots with ≥ 70% composition of pine or spruce or mixed conifer stands with <10% birch or other deciduous trees; poor stocked = poorly stocked stands under regeneration or mature stands with a basal area of maximally 3-5 m 2 ha −1 depending on site index class. Tree composition percentages are based on crown cover for sapling to commercial size trees (Supplementary material, Appendix A) and on volume in commercial sized trees.

Component birch biomass functions
A summary of selected sample tree, stand, site characteristics, and climate data is presented in Table 1. The sampled trees in the current study were representative of the dbh and height ranges of the birch trees recorded in the 9th NNFI (2005)(2006)(2007)(2008)(2009) in Norway. The sampled dbh range was 4.0-45.5 cm and the NNFI range was 5.0-69.1 cm with only 38 trees with diameters larger than the largest tree in the study. The sampled height range was 5.8-29.6 m and the NNFI range was 1.7-30.8 m with only one tree taller than the tallest tree in the study.
Separate functions were derived using the predictors dbh and height for total aboveground (TAG d , TAG dh ), total stem (TS d , TS dh ), stemwood (SW d , SW dh ), stem bark (SB d , SB dh ), live crown (LC d , LC dh ), live branch (LB d , LB dh ), leaf (LF d ), and dead branch (DB d ) birch biomass (Tables 2 and 3). There were no trends in the Pearson residuals across the range of the predicted response for any of the functions (including those not shown), with the possible exception of DB d (Figure 2H), indicating that the individual function fits were good (Figure 2).
Incorporating height into the single-variable functions reduced function error (RMSE) markedly for TAG (18.0%), TS (55.4%), SW (66.4%), LC (12.2%), and LB (16.4%) biomass, but did not improve SB (−44.9%) biomass (Tables 2 and 3). No convergent functions were found by including height with the LF d and DB d biomass functions. The best suite of aboveground birch biomass component functions in the current study by RMSE was: TAG dh , TS dh , SW dh , SB d , LC dh , LB dh , LF d , and DB d (Tables 2 and 3; Figure 2).

Comparing the functions with existing data
Applying the derived total aboveground function (TAG dh ) to Norwegian total aboveground mountain birch data inventoried by Bollandsås et al. (2009), resulted in an underestimation of the measured biomass by 5.3 kg (p = 0.0083). The derived total aboveground biomass function also showed a trend to especially underestimate trees with a small dbh ( Figure 3A). A weak trend (p = 0.0072) to overestimate biomass by increasing dbh was indicated; however, this trend was solely due to two large influential observations ( Figure  3A). The total stem biomass function was also found to underestimate by 3.8 kg (p < 0.0001) across the range of dbh (data not shown). A trend was found to overestimate total crown biomass in trees with dbh greater than 6 cm (p = 0.0001) ( Figure 3B). The derived complete crown (LC dh + DB d ) biomass functions were also found to overestimate by 9.7 kg (p = 0.0036) across the range of dbh.
The derived functions were also applied to a regional dataset inventoried in Sweden (Marklund 1987(Marklund , 1988). The derived functions for different aboveground components significantly underestimated the measured biomass by: SW dh (9.6 kg), SB d (5.9 kg), LB dh (11.4 kg), and DB d (1.3 kg). For each of the functions (SB d , LB dh , and DB d ) except SW dh , a weak but significant trend to underestimate by increasing dbh was found (Figure 4). When the prediction errors were fit with height as the independent variable, the derived functions showed a significant trend to underestimate Swedish birch biomass for SW dh , SB d , LB dh , and DB d (data not shown) with increasing dbh. With crown height as the independent variable, the prediction errors showed a similar significant trend for the functions SW dh and SB d (data not shown). No trends were found with respect to age at breast height, site index, or elevation.

Norwegian birch biomass stock and stock change estimates
The derived functions predicted 2.2% and 14.3% higher total birch biomass stock estimates (86.3 million tons) for NNFI9 than when using the Marklund (1987Marklund ( , 1988 and Bollandsås et al. (2009) functions, respectively ( Figure 5A). The derived functions also predicted 0.53 and 2.04 million tons higher biomass stock change (6.6 million tons) than when using the corresponding functions ( Figure 5A). The stock estimate was more sensitive to biomass function errors than the stock change estimates. The relative differences between the stock and stock change predictions were similar and consistent for most comparisons (Figure 5), so further descriptions of the comparisons will be in terms of the stock estimates alone.
When the different function estimates were stratified by region, site productivity, and forest type, several trends became evident. The derived function predictions differed markedly compared to those obtained with the functions of Bollandsås across all stratifications ( Figure 5A-C). Regional trends revealed that the derived function predictions were higher than those obtained with the functions of Marklund in the west (5.0%), less so in central and northern Norway, and nearly the same in the southeast (0.2%) ( Figure 5A). The derived functions predicted much higher in the southeast and west and less so in central and northern Norway compared to Bollandsås ( Figure 5A). Compared to Marklund, the derived functions predicted higher on unproductive sites (10.4%), less so on low productive sites, nearly the same on medium sites, and lower (−3.1%) on highly productive sites ( Figure 5B). Conversely, the derived functions predicted lower than Bollandsås on unproductive sites (−6.0%), but increasingly higher on low, medium, and high site productivity classes ( Figure 5B). Grouping by forest type showed that the derived functions predicted higher (ca. 3.5%) than  (Marklund 1987(Marklund , 1988. Predicted values are calculated from stemwood (SW dh ), stem bark (SB d ), live branch (LB dh ), and dead branch (DB d ) biomass functions from the current study.
Marklund in birch dominant, other deciduous, and poor stocked forest types, but nearly the same in mixed and conifer dominant types ( Figure 5C). Compared to Bollandsås, the derived functions predicted 11.2-21.8% higher depending on the forest type, with the least difference in the birch dominant type ( Figure 5C).  ; site productivity (B); and forest type (C). The zero line represents the total aboveground birch biomass estimate of the current study (TAG S ) for all graphs. Bars for the stock estimates (left side) depict the percent difference of the current study estimates from those of Marklund (1987Marklund ( , 1988 or Bollandsås et al. (2009), respectively. Bars for the stock change estimates (right side) depict the difference in stock change estimates expressed in million tons biomass (m.t.b.) of the current study from those of Marklund (1987Marklund ( , 1988

Discussion
The presented results suggest that the derived functions are the best available for regional birch biomass stock and stock change estimation in Norway. The derived functions provide a good fit to the data with no visible trends in the residuals for the most important aboveground biomass components (Figure 2). Predictions obtained with the new functions showed good predictive ability for Norwegian mountain birch and regional Swedish birch biomass data. However, the observed underestimation patterns suggest that the biomass allocation pattern of the respective birch populations were different (Figures 3 and 4). The derived functions mostly estimated higher birch biomass stock and stock change throughout Norway, across different regions, site productivities, and forest types ( Figure 5) than did existing Norwegian mountain birch or Swedish functions. The 67 birch trees were sampled from throughout much of the birch zone in Norway, covering areas not previously represented by the existing local Norwegian birch functions including: low-elevation western coastal, mid-elevation southeastern, central, and northern Norway ( Figure 1). The majority of the most prevalent conditions in which birch occurs in Norway (Table 1) were also represented in the sample; however, unproductive forest, high elevation birch in southern Norway (>700 meters above sea level), and birch growing on peatlands were not included. There are considerable land areas in northern Norway and at high elevations throughout Norway with unproductive birch forests; while relatively little birch is found on peatlands (NNFI 2009). Therefore, special care should be taken when applying the derived functions to these forest types.
Even though the sample did not include unproductive forests, the sample did include individual trees from very low productive areas with high similarity to much of Norway's unproductive forests. The sample contained 12 trees from >540 m.a.s.l. from the west and southeast as well as 12 trees from >180 m.a.s.l. in the north. Environmental conditions on these sample sites are approximately similar to the conditions found on birchdominated unproductive forest in the north and at high elevations throughout Norway. Some indication of the expected performance of the derived functions on unproductive sites is indicated in the stock and stock change comparison, where the estimate was intermediate to the Bollandsås et al. (2009) and Marklund (1987, 1988) estimates ( Figure 5B). Although it is likely that the derived functions will underestimate southern mountain birch biomass when applied in those conditions, they produce less of an underestimate than the Marklund functions on these sites. It is also important that the data material used for developing the southern Norwegian mountain birch functions (Bollandsås et al. 2009) is limited to three sample areas and does not encompass unproductive birch elsewhere in Norway (Bollandsås et al. 2009).
Geographically, extrapolating allometric biomass functions is common practice in regions where no functions exist. Available evidence suggests that this practice can have varying effects on the estimation of component birch biomass. In a widely distributed genus such as birch, growing conditions can range from similar to dissimilar in different regions likely resulting in increasingly different biomass allocation patterns where conditions are most dissimilar. This hypothesis has circumstantial support from comparative Nordic allometric birch biomass studies which have reported both differing and similar birch biomass component estimates. Bollandsås et al. (2009) found various significant differences from measured Norwegian mountain birch biomass compared with predicted values for stem, total crown, and total aboveground biomass using a suite of local and regional birch biomass functions from Norway (Opdahl 1987;Korsmo 1995) and Sweden (Marklund 1987(Marklund , 1988Bylund & Nordell 2001;Claesson et al. 2001;Dahlberg et al. 2004). Bollandsås et al. (2009) reported no significant difference for total aboveground or stem biomass compared to the regional Icelandic functions of Snorrason and Einarsson (2006). Repola (2008) reported differing biomass predictions for birch live branch biomass with increasing dbh in a comparison of his and Marklund's (1987Marklund's ( , 1988 functions applied to Finnish NFI data, but relatively similar stem biomass estimates. In the current study, NNFI total aboveground biomass stock estimates were the same as Marklund's (1987Marklund's ( , 1988 estimate in southeastern Norway where conditions are most similar to Swedish conditions, but increasingly different in northern, central, and western Norway ( Figure 5A) where conditions are most dissimilar. Component birch biomass was also significantly different than southern Norwegian mountain birch ( Figure 3) and Swedish birch (Figure 4).
The derived functions' underestimate of measured Swedish biomass, but higher estimate on NNFI data is likely caused by the large proportion of low and unproductive birch forests in Norway compared to the data sampled by Marklund. For forest with relatively high productivity, the Marklund functions estimate slightly higher biomass than the derived functions while the opposite is the case for low and unproductive forests ( Figure 5B).

Conclusions
The results indicate that geographic extrapolation of birch biomass functions can lead to divergent biomass estimates. Circumstantial evidence from this and other comparative birch biomass studies from the Nordic countries suggest that it is due to differences in biomass allocation patterns in trees growing in different conditions. If the estimated differences presented here are representative of the actual error that would result, then the continued application of Marklund's functions for stock and stock change estimation for carbon accounting and bioenergy stock predictions may result in an underestimation of birch biomass throughout Norway, in the west, in central, and in the north ( Figure  5A). The comparison of the derived functions applied to existing biomass and NNFI data indicates that the functions are likely the best choice for estimating regional birch biomass stock and stock change in Norway.